Summability of sequences and extensions of operator ideals

Abstract

Given a Banach space Y, the so-called right Y and left Y-extensions of operator ideals are introduced and applied to the study of operator ideal properties of well-known classes of operators on Banach spaces. The key is to introduce a general theory (in the context of operator ideal theory) to study some classes of operators which are right or left Y-extensions of classical operator ideals for some Banach space Y. This leads to a study of a class of operators, called (r, Y_q)-limited operators, which generalises some recently introduced operators in the literature. In particular, the results of this study are applied to obtain characterisations and new information about the sequentially p-limited operators, weakly mid (q,p; r)-summing operators, p-convergent (or p-converging) operators and weak* p-convergent operators.

Mathematics Subject Classification (2020):

• 47B10
• 47L20
• 46B45

Keywords:

• Banach operator ideal
• right-Y-extension
• left-Y-extension
• [Y, p]-summable sequence
• (r, Y_q)-limited operator